There is a myth that circulates among beginning calculusstudents which states that all

Chapter 3, Problem 58

(choose chapter or problem)

There is a myth that circulates among beginning calculus students which states that all indeterminate forms of types \(0^{0}\), \(\infty^{0}\), and \(1^{\infty}\) have value 1 because “anything to the zero power is 1” and “1 to any power is 1.” The fallacy is that \(0^{0}\), \(\infty^{0}\), and \(1^{\infty}\) are not powers of numbers, but rather descriptions of limits. The following examples, which were suggested by Prof. Jack Staib of Drexel University, show that such indeterminate forms can have any positive real value:

(a) \(\lim _{x \rightarrow 0^{+}}\left[x^{(\ln a) /(1+\ln x)}\right]=a\left(\text { form } 0^{0}\right)\)

(b) \(\lim _{x \rightarrow+\infty}\left[x^{(\ln a) /(1+\ln x)}\right]=a\left(\text { form } \infty^{0}\right)\)

(c)  \(\lim _{x \rightarrow 0}\left[(x+1)^{(\ln a) / x}\right]=a\left(\text { form } 1^{\infty}\right)\)

Verify these results.

Equation Transcription:

Text Transcription:

0^0

infinity^0

1^infinity

0^0

infinity^0

1^infinity

lim _x right arrow 0^+ [x^(ln a)/(1+ln x) ]=a(form 0^0)

lim _x right arrow + infinity [x^(ln a)/(1+ln x) ]=a(form infinity^0 )

lim _x right arrow 0 [(x+1)^(ln a)/x ]=a(form 1^infinity)